Finite Group DatabasebetaFinite Group Databasebeta| order | 729 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of H(3,𝔽3) |
| automorphism | ? |
| homology | Abelianization (H1) = C33×C9 · Schur multiplier (H2) = C35 |
| direct product | C3×#6903 |
| subgroup | G/H | |
|---|---|---|
| center | C3×C27 | C32 |
| commutator | C3 | C33×C9 |
| Frattini, cosocle | C9 | C34 |
| socle | C32 | C32×C9 |
| prime radical | C3×H(3,𝔽3) | C9 |
| LCS | G ▷ C3 ▷ C1 | |
| UCS | C1 ◁ C3×C27 ◁ G | |
| LECS | G ▷ C9 ▷ C3 ▷ C1 | |
424 conjugacy classes of subgroups.
| minimal degree | μ(G) = 84 |
| transitive degree | μt(G) = 243 |
A minimal faithful permutation representation — 4 generators on 84 points:
(1,2,3);(4,5,9,7,11,21,17,27,42,8,12,22,18,28,43,36,51,65,19,29,44,37,52,66,58,72,80)(6,10,20,15,25,40,34,49,63,16,26,41,35,50,64,57,71,79,14,24,39,33,48,62,56,70,78)(13,23,38,31,46,60,54,68,76,30,45,59,53,67,75,73,81,83,32,47,61,55,69,77,74,82,84);(4,6,13)(5,10,23)(7,15,31)(8,16,30)(9,20,38)(11,25,46)(12,26,45)(14,32,19)(17,34,54)(18,35,53)(21,40,60)(22,41,59)(24,47,29)(27,49,68)(28,50,67)(33,55,37)(36,57,73)(39,61,44)(42,63,76)(43,64,75)(48,69,52)(51,71,81)(56,74,58)(62,77,66)(65,79,83)(70,82,72)(78,84,80);(6,14,16)(10,24,26)(13,30,32)(15,33,35)(20,39,41)(23,45,47)(25,48,50)(31,53,55)(34,56,57)(38,59,61)(40,62,64)(46,67,69)(49,70,71)(54,73,74)(60,75,77)(63,78,79)(68,81,82)(76,83,84)297 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(729, 477);